Table Of ContentDomains and Event Structures for Fusions
Paolo Baldan
University of Padova
Andrea Corradini, Fabio Gadducci
University of Pisa
Abstract—Stableeventstructures,andtheirdualitywithprime {a,b,c}
algebraic domains (arising as partial orders of configurations),
{a,b} {a,c}
are a landmark of concurrency theory, providing a clear char-
acterisation of causality in computations. They have been used {a} {c}
7 for defining a concurrent semantics of several formalisms, from
1 Petri nets to linear graph rewriting systems, which in turn lay ∅
0 at the basis of many visual frameworks. Stability however is
2 restrictive for dealing with formalisms where a computational Fig. 1: The domain of configurations of the process a.b|c.
step can merge parts of the state, like graph rewriting systems
n with non-linear rules, which are needed to cover some relevant
a applications(suchasthegraphicalencodingofcalculiwithname
J
passing). We characterise, as a natural generalisation of prime
a transition of a CCS process or a transition firing for a Petri
9 algebraicdomains,aclassofdomainsthatiswell-suitedtomodel
the semantics of formalisms with fusions. We then identify a net)withallitscauses.Asasimpleexample,considertheCCS
] corresponding class of event structures, that we call connected process a.b|c. The corresponding partial order is depicted in
O
eventstructures,viaadualityresultformalisedasanequivalence Fig.1.Theeventscorrespondtoconfigurations{a}(transition
L ofcategories.Connectedeventstructuresareexactlytheclassof a with empty set of causes), {a,b} (transition b caused by a)
s. erevwenrtitisntrguscytustreemsst.hIentaerriessetiansgltyh,ethseemcaatnetgiocrsyooffngoenn-elirnaelaurngstraabplhe and {c} (transition a with empty set of causes). The fact that
c event structures coreflects into our category of domains, so that eacheventinaconfigurationhasauniquelydeterminedsetof
[
our result provides a characterisation of the partial orders of causes, a property that for event structures is called stability,
1 configurations of such event structures. allows one to characterise such elements, order theoretically,
v Index Terms—Event structures, fusions, graph rewriting, pro- astheprimeelements:iftheyareincludedinajointheymust
4 cess calculi.
beincludedinoneofthejoinedelements.Eachelementofthe
9
partial order of configurations can be reconstructed uniquely
3
2 I. INTRODUCTION as the join of the primes so that the partial order is prime
0 algebraic. This duality between event structures and domains
Since a long time stable/prime event structures and their
1. duality with prime algebraic domains have been considered of configurations can be nicely formalised in terms of an
0 one of the landmarks of concurrency theory, providing a clear equivalence between the category of prime event structures
7 and that of prime algebraic domains [1], [15].
characterisation of causality in software systems. They have
1
been used to provide a concurrent semantics to a wide range Thesetupdescribedsofarfailswhenmovingtoformalisms
:
v of fundational formalisms, from Petri nets [1] to linear graph where a computational step can merge parts of the state, as
i
X rewriting systems [2]–[4] and process calculi [5]–[7]. They it happens whenever we consider nominal calculi where as a
are one of the standard tools for the formal treatment of (true, result of name passing the received name is identified with
r
a i.e.,non-interleaving)concurrency.See,e.g.,[8]forareasoned a local one at the receiver [16], [17] or in the modelling of
survey on the use of such causal models. Recently, they bonding in biological/chemical processes [18]. Whenever we
have been used in the study of concurrency in weak memory think of the state of the system as some kind of graph with
models[9],[10]andforprocessmininganddifferencing[11]: the dynamics described by graph rewriting, this means that
In order to endow a chosen formalism with an event struc- rulesarenon-linear(precisely,intheso-calleddoublepushout
turesemantics,astandardconstructionconsistsinviewingthe approach [19], left-linear but possibly not right-linear). In
class of computations as a partial order. An elements of the general terms, the point is that in the presence of fusions the
order is some sort of configuration, i.e., an execution trace sameeventcanbeenabledbydifferentminimalsetsofevents,
up to an equivalence that identifies traces differing only for thus preventing the identification of a notion of causality.
the order of independent steps (e.g., interchange law [12] AsanexampleconsiderthegraphrewritingsysteminFig.2.
in term rewriting, shift equivalence [13] in graph rewriting, Figure 2a reports the start graph G and the rewriting rules
s
permutation equivalence [14] in the lambda-calculus, ...), p , p , and p . Observe that rules p , where y can be either
a b c y
and the order relates two computations when the latter is a or b, delete edge y¯ and merge nodes c and ν. The possible
an extension of the former. Events are then identified with rewrites are reported in Fig. 2b. For instance, applying p to
a
configurationsconsistingofamaximalcomputationstep(e.g., G we get the graph G . Now, p can still be applied to G
s b b b
a¯ ν¯ y¯ ν¯ ν¯ ν¯ ν¯ in this context an event is still a computation that cannot be
¯b in py in pc decomposed as the join of other computations hence, in order
theoretical terms, it is an irreducible. However, due to unsta-
c ν c ν c,ν c,ν c,ν bility, irreducibles are not primes: two different irreducibles
can be different minimal histories of the same event, in a way
(a) The start graph G and the rules p (y∈{a,b}) and p
s y c that an irreducible can be included in a computation that is
Ga a¯ ν¯ in pc Gac a¯ ν¯ Gc tohfethjoeintwoof.tFwoor cinosmtapnuctea,tiionnsthweiethxoaumtpbleeinagboivnec,lu{dae,dc}inisanayn
pb c,v pa ν¯ in Gab c,ν pa ν¯ irreducible, corresponding to the execution of c enabled by
G pc a, and it is included in {a} (cid:116) {b,c} = {a,b,c}, although
s neither {a,c} ⊆ {a} nor {a,c} ⊆ {b,c}. Uniqueness of
pa ¯b ν¯ in pb c,ν ¯b ν¯ pb c,ν decompositionofanelementintermsofirreduciblesalsofails,
e.g., {a,b,c} = {a}(cid:116){b}(cid:116){a,c} = {a}(cid:116){b}(cid:116){b,c}: the
pc
Gb c,v Gbc c,ν irreducibles {a,c} and {b,c} can be used interchangeably.
Building on the previous observation, we introduce an
(b) The possible rewrites equivalence on irreducibles identifying those that can be
{a,b,c} used interchangeably in the decompositions of an element
(intuitively, different minimal histories of the same event).
{a,c} {b,c}
Based on this we give a weaker notion of primality (i.e., up
{a,b}
to interchangeability) such that the class of domains suited
{a} {b}
for modelling the semantics of formalisms with fusions are
∅ defined as the class of weak prime algebraic domains.
Given a weak prime algebraic domain, a corresponding
(c) The domain of configurations
event structure can be obtained by taking as events the set
Fig. 2: A graph rewriting system with fusions. of irreducibles, quotiented under the (transitive closure of
the) interchangeability relation. The resulting class of event
structures is a (mild) restriction of the general unstable event
matching its left-hand side non-injectively, thus getting graph
structures in [15] that we call connected event structures.
G . Similarly, we can apply first p and then G , obtaining
ab b a Categorically, we get an equivalence between the category
againG .Observethatatleastonebetweenp andp mustbe
ab a b of weak prime algebraic domains and the one of connected
applied to enable p , since the latter rule requires nodes c and
c event structures, generalising the equivalence between prime
ν to be merged. The graph rewriting system is a (simplified)
algebraic domains and prime event structures.
representationoftheπ-calculusprocess(νc)(a¯(c)|¯b(c)|c()).
We also show that, in the same way as prime algebraic
Rules p , for y ∈{a,b}, represent the execution of y¯(c) that
y domains/prime event structures are exactly what is needed for
outputsonchannelytherestrictednamec.Thefirstrulethatis
Petri nets/linear graph rewriting systems, weak prime alge-
executed extrudes name c, while the second is just a standard
braic domains/connected event structures are exactly what is
output.Onlyaftertheextrusionthenamecisavailableoutside
needed for non-linear graph rewriting systems: each rewriting
the scope and the input prefix c() can be consumed. Observe
systemmapstoaconnectedeventstructureandviceversaeach
that in a situation where all the three rules p , p , and p are
a b c connected event structure arises as the semantics of some
applied, since p and p are independent, it is not possible
a b rewriting system. This supports the adequateness of weak
to define a proper notion of causality. We only know that at
prime algebraic domains and connected event structure as
least one between p and p must be applied before p . The
a b c semantics structures for formalims with fusions.
corresponding domain of configurations, reported in Fig. 2c,
Interestingly, we can also show that the category of general
is naturally derived from the possible rewrites in Fig. 2b.
unstable event structures [15] coreflects into our category of
The impossibility of modelling these situations with stable
weak prime algebraic domains. Therefore our notion of weak
event structures is well-known (see, e.g., [15] for a general
primealgebraicdomaincanbeseenasanovelcharacterisation
discussion, [2] for graph rewriting systems or [16] for the π-
of the partial order of configurations of such event structures
calculus).Onehastodropthestabilityrequirementandreplace
that is alternative to those based on intervals in [20], [21].
causality by an enabling relation (cid:96). More precisely, in the
Our characterisation is a natural generalisation of the one for
specific case we would have ∅(cid:96)a, ∅(cid:96)b, {a}(cid:96)c, {b}(cid:96)c.
prime event structures, with irreducibles (instead of primes)
The questions that we try to answer is: what can be having a tight connection with events. The correspondence
retained of the satisfactory duality between events structures is established at a categorical level, as a coreflection of
and domains, when dealing with formalisms with fusions? categories, something that, to the best of our knowledge, had
Which are the properties of the domain of computations that not been done before in the literature.
ariseinthissetting?Whataretheeventstructurecounterparts? The rest of the paper is structured as follows. In Section II
The domain of configurations of the example suggests that we recall the basics of (prime) event structures and their
correspondence with prime algebraic domains. In Section III The classes of stable and prime es play an important role
we introduce weak prime algebraic domains, connected event in the paper.
structures and establish a duality result. In Section IV we
Definition 4 (stable and prime es) An es (cid:104)E,(cid:96),#(cid:105) is sta-
show the intimate connection between weak prime algebraic
ble if X (cid:96) e, Y (cid:96) e, and X ∪ Y ∪ {e} consistent imply
domains or equivalently connected event structures, and non-
X∩Y (cid:96)e. It is prime if X (cid:96)e and Y (cid:96)e imply X∩Y (cid:96)e.
linear graph rewriting systems. Finally, in Section V we wrap
up the main contributions of the paper and we sketch further For stable es, given a configuration C and an event e∈C,
advances and possible connections with related works. there is a unique minimal configuration C(cid:48) ⊆ C such that
C(cid:48) (cid:96) e. The set C(cid:48) can be seen as the set of causes of
0
II. BACKGROUND:DOMAINSANDEVENTSTRUCTURES the event e in the configuration C. This gives a well-defined
This section recalls the notion of event structures, as intro- notion of causality that is local to each configuration. In a
duced in [15], and their duality with partial orders. prime es, for any event e there is a unique minimal enabling
C (cid:96) e,thusprovidingaglobalnotionofcausality.Ingeneral,
0
A. Event structures
in possibly unstable es, due to the presence of consistent or-
In the paper, we focus on event structures with binary enablings, there might be distinct minimal enablings in the
conflict.Thischoiceplaysaroleonlyintherelationwithgraph same configuration.
rewriting (Section IV), while the duality results in Section III
Example 1 Asimpleexampleofunstableesisthefollowing:
can be easily rephrased for event strutures with non-binary
the set of events is {a,b,c}, the conflict relation # is the
conflicts expressed by means of a consistency predicate (see
empty one and the minimal enablings are ∅ (cid:96) a, ∅ (cid:96) b,
Appendix A). Given a set X we denote by 2X and 2X the 0 0
fin {a} (cid:96) c, and {b} (cid:96) c. Thus, event c has two minimal
powerset and the set of finite subsets of X, respectively. For 0 0
enablingsandtheseareconsistent,hence{a,b}(cid:96)c.Thisisthe
m,n∈N, we denote by [m,n] the set {m,m+1,...,n}.
eventstructurediscussedintheintroduction,whosedomainof
Definition 1 (event structure) An event structure (es for configurations is reported in Fig. 2c.
short) is a tuple (cid:104)E,(cid:96),#(cid:105) such that
The class of es can be turned into a category.
• E is a set of events;
• a(cid:96)n⊆d X2Efin⊆×YEimisplitehseYen(cid:96)abeli;ng relation satisfying X (cid:96) e DEe2fiinsiatiopnart5ial(cfautnecgtoiornyfof:eEs1) →A Em2orspuhcihsmthaotffoesrafll:e1E,1e(cid:48)1→∈
• #⊆E×E is the conflict relation. E1 with f(e1), f(e(cid:48)1) defined
An X ⊆E is consistent if there is no e,e(cid:48) ∈X with (e#e(cid:48)). • if f(e1)#f(e(cid:48)1) then e1#e(cid:48)1
• if f(e1)=f(e(cid:48)1) and e1 (cid:54)=e(cid:48)1 then e1#e(cid:48)1;
An es (cid:104)E,(cid:96),#(cid:105) is often denoted simply by E. Computa- • if X1 (cid:96)1 e1, for X1 ⊆E, then f(X1)(cid:96)2 f(e1).
tions are captured by the notion of configuration.
We denote by ES the category of es and their morphisms and
Definition 2 (configuration, live es) A configuration of an by sES and pES the full subcategories of stable and prime es.
es E is a finite consistent C ⊆ E which is secured, i.e.,
B. Domains
C can be totally ordered as (cid:104)e ,...,e (cid:105) in a way that
1 n Apreorderedorpartiallyorderedset(cid:104)D,(cid:118)(cid:105)isoftendenoted
{e ,...,e } (cid:96) e for all k ∈ [1,n]. The set of config-
1 k−1 k simply as D, omitting the (pre)order relation. Given an ele-
urations of an es E is denoted by Conf(E). An es is live
ment x∈D, we write ↓x to denote the set {y ∈D |y (cid:118)x}.
if conflict is saturated, i.e., for all e,e(cid:48) ∈ E, if there is no
Wedenoteby(cid:22)theimmediatepredecessorrelation,i.e.,x(cid:22)y
C ∈Conf(E) such that {e,e(cid:48)}⊆C then e#e(cid:48) and moreover
when x (cid:118) y and for all z, if x (cid:118) z (cid:118) y then z ∈ {x,y}. A
¬(e#e) for all e∈E.
subset X ⊆ D is consistent if it has an upper bound d ∈ D
Observe that when a configuration C is secured via a total (i.e., x (cid:118) d for all x ∈ X). It is pairwise consistent if each
ordering(cid:104)e ,...,e (cid:105),inparticular,∅(cid:96)e .Inthissetting,two two elements subset Y of X is consistent. A subset X ⊆ D
1 n 1
eventsareconcurrentwhentheyareconsistentandenabledby is directed if X (cid:54)= ∅ and each pair of elements in X has an
the same configuration. upperboundinX.Itisanidealifitisdirectedanddownward
closed. Given D, its ideal completion, denoted Idl(D), is the
Remark 1 Intherestofthepaper,werestricttolivees,where
setofidealsofD,orderedbysubsetinclusion.Theleastupper
conflictissaturated(thiscorrespondstoinheritanceofconflict
bound and the greatest lower bound of a subset X ⊆ D (if
inprimeeventstructures)andeacheventisexecutable.Hence (cid:70)
they exist) are denoted by X and (cid:100)X, respectively.
the qualification live is omitted.
Definition 6 (domains) A partial order D is coherent if for
Sincetheenablingpredicateisoverfinitesetsofevents,we (cid:70)
any pairwise consistent X ⊆ D the least upper bound X
can consider minimal sets of events enabling a given one.
exists. An element d ∈ D is compact if for any directed
(cid:70)
Definition 3 (minimal enabling) Given an es (cid:104)E,(cid:96),#(cid:105) de- X ⊆ D, d (cid:118) X implies d (cid:118) x for some x ∈ X.
fine C (cid:96) e when C ∈ Conf(E), C (cid:96) e and for any other The set of compact elements of D is denoted by K(D). A
0
configuration C(cid:48) ⊆C, if C(cid:48) (cid:96)e then C(cid:48) =C. coherent partial order D is algebraic for any x∈D we have
(cid:70) (cid:70)
x = (↓x∩K(D)). We say that D is finitary if for any pairwise consistent X ⊆ K(D), if i = X then i ∈ X.
element a ∈ K(D) the set ↓a is finite. We refer to algebraic The set of irreducibles of D is denoted by ir(D) and, for
finitary coherent partially ordered sets as domains. d∈D, we define ir(d)=↓d∩ir(D).
For a domain D we can think of its elements as “pieces of Irreducibles in domains have a simple characterisation.
information” expressing the states of evolution of a process.
Lemma 1 (unique predecessor for irreducibles) Let D be
Compact elements represent states that are reached after a
a domain and i ∈ D. Then i ∈ ir(D) iff it has a unique
finite number of steps. Thus algebraicity essentially says that
immediate predecessor, denoted p(i).
any infinite computation can be approximated with arbitrary
precision by the finite ones. More formally, when D is Proof: Assume that i ∈ D has a unique immediate
algebraic it is determined by K(D), i.e., D (cid:39)Idl(K(D)). predecessor d≺i, and let X ⊆K(D) be consistent and such
For an es, the configurations ordered by subset inclusion that i=(cid:70)X. Hence for any x∈X we have x(cid:118)i. Assume
form a domain. When the es is stable, if an event with its by contradiction that i (cid:54)∈ X. This means that all elements
minimal history is in the join of different configurations, then x∈X must be below the immediate predecessor x(cid:118)d, and
itbelongs,withthesamehistory,tooneofsuchconfigurations. therefore i = (cid:70)X (cid:118) d ≺ i, which is a contradiction. Hence
Inorder-theoreticterms,minimalhistoriesareprimeelements, it must be i∈X, which means that i is irreducible.
representing the building blocks of computations.
Vice versa, let i be irreducible and let d ,d ≺ i be
1 2
Definition 7 (primes and prime algebraicity) Let D be a immediate predecessors. Since D is a domain and {d1,d2} is
domain. A prime is an element p ∈ K(D) such that, for any consistent, we can take d=d1(cid:116)d2 and we know d1 (cid:118)d(cid:118)i.
pairwise consistent X ⊆ K(D), if p (cid:118) (cid:70)X then p (cid:118) x for Since i is irreducible it cannot be d = i, therefore d = d1
some x ∈ X. The set of prime elements of D is denoted by and thus d1 =d2. This means that i has a unique immediate
pr(D). The domain D is prime algebraic (or simply prime) predecessor.
if for all x∈D we have x=(cid:70)(↓x∩pr(D)). We next observe that any domain is actually irreducible
algebraic, namely it can be generated by the irreducibles.
Prime domains are the domain theoretical counterpart of
stable and prime es. For a stable es (cid:104)E,#,(cid:96)(cid:105), the ideal com- Proposition 1 (domains are irreducible algebraic) Let D
(cid:70)
pletion of (cid:104)Conf(E),⊆(cid:105) is a prime domain, denoted D (E). be a domain. Then for any d∈D it holds d= ir(d).
S
Viceversa, given a prime domain D, the triple (cid:104)pr(D),#,(cid:96)(cid:105),
Proof: We first prove that for any compact element
where p#p(cid:48) if {p,p(cid:48)} is not consistent and X (cid:96) p when
(cid:70)
d ∈ K(D) it holds that d = (↓d∩ir(D)). The thesis then
(↓p∩pr(D))\{p}⊆X, is a prime es, denoted E (D).
S immediately follows from algebraicity. Since D is a domain,
This correspondence can be elegantly formulated at cate-
↓d is finite, hence we can proceed by induction on |↓d|.
gorical level. We recall the notion of domain morphism [15].
When |↓d|=1, we have that d=⊥, hence ↓d∩ir(D)=∅
Definition 8 (category of prime domains) Let D1, D2 be and indeed ⊥ = (cid:70)∅. When |↓d| = k > 1 consider the
prime domains. A morphism f :D1 →D2 is a total function immediatepredecessorsofdanddenotethemd1,...,dn ≺d.
such that Since D is a domain and {d ,...,d } is consistent, there
1 n
1) d1 (cid:22)d(cid:48)1 implies f(d1)(cid:22)f(d(cid:48)1); exists (cid:70){d1,...,dn} = d(cid:48) and di (cid:118) d(cid:48) (cid:118) d. There are two
2) for X ⊆D consistent, f((cid:70)X )=(cid:70)f(X ); cases
1 1 1 1
3) for X1 ⊆D1 consistent, f((cid:100)X1)=(cid:100)f(X1); • d(cid:48) =di, for all i∈[1,n], i.e., d has a unique immediate
We denote by pDom the category of prime domains and their predecessor, hence it is an irreducible and thus clearly
morphisms. d=(cid:70)(↓d∩ir(D)) or
Theorem 1 (duality) There are functors D : sES → pDom • d = d(cid:48) = (cid:70){d1,...,dn}. Since, in turn, by inductive
S hypothesis d = (cid:70)(↓d ∩ir(D)) and ↓d∩ir(D) =
and E :pDom→sES establishing a coreflection. It restricts i i
S (cid:83)n (↓d ∩ir(D)), we immediately get the thesis.
to an equivalence of categories between pDom and pES. i=1 i
III. WEAKPRIMEDOMAINSANDCONNECTEDES
Now note that any prime is an irreducible. If D is a prime
In this section we characterise a class of domains, and the domain then also the converse holds. i.e., the irreducibles
corresponding brand of es, that are suited for expressing the coincide with the primes.
semantics of computational formalisms with fusions.
Proposition 2 (irreducibles vs. primes) LetD beadomain.
A. Weak prime algebraic domains Then D is a prime domain iff pr(D)=ir(D).
We show that domains arising in the presence of fusions
Proof: Let D be a prime domains. We observed that
are characterised by resorting to a weakened notion of prime
pr(D) ⊆ ir(D) holds in general domains. For the con-
element. We start recalling the notion of irreducible element.
verse inclusion, let i ∈ ir(D). By prime algebraicity
(cid:70)
Definition 9 (irreducibles) Let D a be a domain. An irre- i = ↓i∩pr(D). Since i is irreducible, there exists p ∈
ducible of D is an element i ∈ K(D) such that, for any ↓i∩pr(D) such that i=p, hence i is a prime.
Vice versa, if D is a domain, by Proposition 1 we know downward closed, and (cid:70)(X∪{i})=(cid:70)(X∪{i(cid:48)}). Therefore
that D is irreducible algebraic. Hence, if pr(D)=ir(D), we i,i(cid:48) (cid:118)(cid:70)(X∪{i})=(cid:70)(X∪{i(cid:48)})andthusi,i(cid:48) areconsistent.
immediately conclude that D is prime. For the second part, let i (cid:118) i(cid:48). If i (cid:54)= i(cid:48) and we let
Quite intuitively, in the domain of configurations of an es X = ir(p(i(cid:48))), it turns out that X ∪{i} = X and X ∪{i(cid:48)}
(cid:70)
the irreducibles are minimal histories of events. For instance, are consistent and downward closed. Moreover X ∪{i}=
in the domain depicted in Fig. 2c the irreducibles are {a}, (cid:70)XX =p(i(cid:48))(cid:54)=(cid:70)X∪{i(cid:48)}=i(cid:48), contradicting i↔i(cid:48).
{b}, {a,c}, and {b,c}. For stable es, the domain is prime and We now give some equivalent characterisations of inter-
thus, as observed above, irreducibles coincide with primes. changeability.
Thisfailsinunstablees,aswecanseeinourrunningexample:
Lemma 4 (characterising ↔) LetDbeadomainandi,i(cid:48) ∈
while {a} and {b} are primes, the two minimal histories of c,
ir(D). Then the following are equivalent
namely{a,c}and{b,c},arenot.Infact,{a,c}⊆{a}(cid:116){b,c},
1) i↔i(cid:48);
but neither {a,c}⊆{a} nor {a,c}⊆{b,c}.
2) i,i(cid:48) consistent and for all d ∈ K(D) such that
The key observation is that in general an event corresponds
p(i),p(i(cid:48)) (cid:118) d, d(cid:116)i = d(cid:116)i(cid:48) and for some such d it
to a class of irreducibles, like {a,c} and {b,c} in our
holds d(cid:54)=d(cid:116)i;
example. Additionally, two irreducibles corresponding to the
3) i,i(cid:48) consistent and i(cid:116)p(i(cid:48))=p(i)(cid:116)i(cid:48) (cid:54)=p(i)(cid:116)p(i(cid:48)).
same event can be used, to a certain extent, interchangeably
for building the same configuration. For instance, {a,b,c} = Proof:
{a,b} ∪ {a,c} = {a,b} ∪ {b,c}. We next formalise this (1 → 2) Assume that i ↔ i(cid:48) and let d ∈ K(D) such that
intuition,i.e.,weinterpretirreduciblesinadomainasminimal p(i),p(i(cid:48))(cid:118)d.IfweletX =ir(d)wehavethatir(i)\{i}⊆
historiesofsomeeventandweidentifyclassesofirreducibles X and similarly ir(i(cid:48))\{i(cid:48)} ⊆ X. Therefore X ∪ {i} and
corresponding to the same event. X∪{i(cid:48)} are downward closed and consistent. Hence d(cid:116)i=
We start by observing that in a prime domain any element (cid:70)X(cid:116)i=(cid:70)(X∪{i})=(cid:70)(X∪{i(cid:48)})=(cid:70)X(cid:116)i(cid:48) =d(cid:116)i(cid:48).
admits a unique decomposition in terms of irreducibles. Moreover, if we consider the set X ⊆ ir(D) required by
(cid:70) (cid:70)
Lemma 2 (unique decomposition) Let D be a prime do- the definition of interchangeability, such that X (cid:54)= (X ∪
main. If X,X(cid:48) ⊆ ir(D) are downward closed sets of irre- {i}) = (cid:70)(X ∪{i(cid:48)}) and define d = (cid:70)X, we obtain d (cid:54)=
(cid:70) (cid:70)
ducibles such that (cid:70)X =(cid:70)X(cid:48) then X =X(cid:48). (X∪{i})= X(cid:116)i=d(cid:116)i, as desired.
Proof: Let X,X(cid:48) ⊆ ir(D) be downward closed sets of (2 → 3) Assume (2). Let p = p(i) (cid:116) p(i(cid:48)). Clearly,
irreducibles such that (cid:70)X = (cid:70)X(cid:48). Take any i(cid:48) ∈ X(cid:48). Then p(i),p(i(cid:48)) (cid:118) p. Therefore i (cid:116) p(i(cid:48)) = i (cid:116) p(i) (cid:116) p(i(cid:48)) =
i(cid:48) (cid:118)x=(cid:70)X. Since the domain is prime algebraic, and thus i(cid:116)p=p(cid:116)i(cid:48) =p(i)(cid:116)p(i(cid:48))(cid:116)i(cid:48) =p(i)(cid:116)i(cid:48).
i(cid:48) is prime, there must exist i ∈ X such that i(cid:48) (cid:118) i and thus Moreover, p(i)(cid:116)p(i(cid:48)) (cid:54)= p(i)(cid:116)i(cid:48), otherwise for any d ∈
i(cid:48) ∈ X. Therefore X(cid:48) ⊆ X. By symmetry also the converse K(D) such that p(i),p(i(cid:48)) (cid:118) d, we would have d (cid:118) d(cid:116)i =
inclusion holds, whence equality. d(cid:116)p(i(cid:48))(cid:116)i=d(cid:116)p(i)(cid:116)p(i(cid:48)), contradicting the second part
The result above no longer holds in domains arising in
of condition (2).
the presence of fusions. For instance, in the domain in
Fig. 2c, X = {{a},{a,c},{b}}, X(cid:48) = {{a},{b},{b,c}}
(3 → 1) Assume (3). Let X ⊆ ir(D) be such that X ∪
and X(cid:48)(cid:48) = {{a},{b},{b,c},{a,c}} are all decompositions {i} and X ∪ {i(cid:48)} are downward closed and consistent sets
for {a,b,c}. The idea is to identify irreducibles that can be
of irreducibles. This implies that ir(p(i)) ⊆ X and similarly
used interchangeably in a decomposition. ir(p(i(cid:48)))⊆X.Hence,ifweletP =(↓p(i)∪↓p(i(cid:48)))∩ir(D)
Definition 10 (interchangeability) Let D be a domain and P ⊆X and (cid:70)P =p(i)(cid:116)p(i(cid:48))
i,i(cid:48) ∈ir(D). We write i↔i(cid:48) if for all X ⊆ir(D) such that
X ∪{i} and X ∪{i(cid:48)} are downward closed and consistent Therefore
we have (cid:70)(X∪{i})=(cid:70)(X∪{i(cid:48)}) and for some such X it (cid:70)(X∪{i})=
holds (cid:70)X (cid:54)=(cid:70)(X∪{i}). =((cid:70)X\P)(cid:116)(cid:70)P (cid:116)i=
=((cid:70)X\P)(cid:116)p(i)(cid:116)p(i(cid:48))(cid:116)i=
Inwords,i↔i(cid:48) meansthatiandi(cid:48) producethesameeffect
=((cid:70)X\P)(cid:116)i(cid:116)p(i(cid:48))=
when added to a decomposition that already includes their
=((cid:70)X\P)(cid:116)p(i)(cid:116)i(cid:48) =
predecessors and there is at least one situation in which the
=((cid:70)X\P)(cid:116)p(i)(cid:116)p(i(cid:48))(cid:116)i(cid:48) =
addition of i and i(cid:48) produces some effect. Hence, intuitively,
=((cid:70)X\P)(cid:116)(cid:70)P (cid:116)i(cid:48) =
i and i(cid:48) correspond to the execution of the same event.
=(cid:70)(X∪{i(cid:48)})
Clearly, interchangeable irreducibles need to be consistent.
Moreover, if we take X = ir(p(i))∪ir(p(i(cid:48))) then X ∪{i}
Lemma 3 Let D be a domain and i,i(cid:48) ∈ ir(D) such that
and X∪{i(cid:48)} are downward closed and consistent. Moreover,
i↔i(cid:48). Then i and i(cid:48) are consistent. And if i(cid:118)i(cid:48) then i=i(cid:48). (cid:70)X =p(i)(cid:116)p(i(cid:48))(cid:54)=p(i)(cid:116)i(cid:48) =(cid:116)(X∪{i(cid:48)}), as desired.
Proof: The fist part is obvious, since, by definition, there The interchangeability relation is reflexive and symmetric,
must exist X ⊆ ir(D) such that X ∪ {i} and X ∪ {i(cid:48)} but not transitive: in the domain of Fig. 3, i↔i(cid:48) and i(cid:48) ↔i(cid:48)(cid:48)
(cid:62) A domain is often built as the ideal completion of its
• • compact elements. We next provide a characterisation of
domains and weak prime domains based on the generators.
i i i
1 • 2 • 3
Lemma 6 (weak prime domains from generators)
p(i ) p(i ) p(i )
1 2 3 Let (P,(cid:118)) be a finitary partial order such that for all
d,d(cid:48),d(cid:48)(cid:48) ∈ P, if {d,d(cid:48),d(cid:48)(cid:48)} is pairwise consistent then d(cid:116)d(cid:48)
⊥
exists and is consistent with d(cid:48)(cid:48). Then Idl(P) is a domain
with K(Idl(P))={↓d|d∈P}(cid:39)P.
Fig. 3: Interchangeability need not be transitive.
Additionally, let ↔ be transitive on consistent irreducibles
and for all i∈ir(P), d,d(cid:48) ∈P consistent, if i(cid:118)d(cid:116)d(cid:48) then
there is i(cid:48) ∈ ir(P), i ↔ i(cid:48) such that i(cid:48) (cid:118) d or i(cid:48) (cid:118) d(cid:48). Then
but not i ↔ i(cid:48)(cid:48). The same holds in the domain obtained by
Idl(P) is a weak prime domain.
removing the top element. We see later that on weak prime
algebraic domains ↔ is transitive on consistent irreducibles Proof: Let (P,(cid:118)) be a finitary partial order such that for
(see Lemma 10). alld,d(cid:48),d(cid:48) ∈P(cid:48),if{d,d(cid:48),d(cid:48)(cid:48)}ispairwiseconsistentthend(cid:116)d(cid:48)
Wenowintroduceweakprimes:theyweakenthepropertyof exists and is consistent with d(cid:48)(cid:48).
primeelements,requiringthatitholdsuptointerchangeability. The fact that Idl(P) is a complete partial order with
K(Idl(P)) = {↓d | d ∈ P} (cid:39) P is a standard result.
Definition 11 (weak prime) Let D be a domain. A weak
Moreover, let X ⊆ Idl(P) pairwise consistent. Consider
prime of D is an element i ∈ ir(D) such that for any (cid:83)
A= {I |I ∈X}. Observe that for any finite Y ⊆A there
consistentX ⊆D,ifi(cid:118)(cid:70)X thenthereexisti(cid:48) ∈ir(D)with (cid:70)
exists Y in P. In fact, let Y = {y ,...,y }. This means
i ↔ i(cid:48) and d ∈ X such that i(cid:48) (cid:118) d. We denote by wpr(D) 1 n
that there are I ,...,I such that y ∈I for each i∈[1,n].
1 n i i
the set of weak primes of D.
Since X is pairwise consistent in Idl(P), we deduce that Y is
Clearly, any prime is a weak prime since interchangeability pairwise consistent in P. Since y1,y2 are consistent, and both
is reflexive. Moreover, in prime domains interchangeability is are consistent with y3,...,yn, by (2) there exists y1(cid:116)y2 and
the identity and also the converse holds. it is consistent with y3,...,yn, i.e., {y1 (cid:116)y2,y3,...,yn} is
again pairwise consistent. Iterating the reasoning we get the
Lemma 5 (weak primes in prime domains) Let D be a (cid:70)
existence of y (cid:116)y (cid:116)...(cid:116)y = Y, as desired. Now, if
1 2 n
prime domain. Then ↔ is the identity and wpr(D)=pr(D). we define I(cid:48) = {(cid:70)Y | Y ⊆ A}, then I(cid:48) is an ideal and
fin
Proof: Let i,i(cid:48) ∈ir(D) be such that i↔i(cid:48). I(cid:48) =(cid:70)X.
Ifiandi(cid:48) arecomparable,i.e.,i(cid:118)i(cid:48) ori(cid:48) (cid:118)i,byLemma3,
we deduce i=i(cid:48) and we are done. For the second part, we need to show that under the
Otherwise, let X = (ir(i)\{i})∪(ir(i(cid:48))\{i(cid:48)}). Note that hypotheses, if I ∈ ir(Idl(P)) and X ⊆ Idl(P) pairwise
X∪{i}andX∪{i(cid:48)}areconsistent,sincebyLemma3,iand consistent and I ⊆ (cid:70)X then there exists I(cid:48) ↔ I and
i(cid:48) areso.MoreoverX∪{i}andX∪{i(cid:48)}aredownwardclosed, A ∈ X such that I(cid:48) ⊆ A. It is immediate to see that
andthus,sincei↔i(cid:48),wededuce(cid:70)(X∪{i})=(cid:70)(X∪{i(cid:48)}). ir(Idl(P)) = {↓i | i ∈ ir(P)}. Thus let I = ir(i) for
(cid:70) (cid:70)
Since D is prime, by Lemma 2, this implies that X ∪{i} = some i ∈ ir(P). The fact that I ⊆ X = {↓d |
(cid:83) (cid:70)
X∪{i(cid:48)}. Since i and i(cid:48) are uncomparable, i,i(cid:48) (cid:54)∈X and thus d ∈ X} means that i (cid:118) {d1,...,dn} for some finite
(cid:83) (cid:70)
we conclude i=i(cid:48). subset {d1,...,dn} ⊆ X. Since i (cid:118) d1 (cid:116) {d2,...,dn},
We argue that the domain of configurations arising in the by the hypothesis there is i1 ↔ i such that i1 (cid:118) d1 or
(cid:70) (cid:70)
presence of fusions can be characterised domain-theoretically i1 (cid:118) {d2,...,dn} = d2 (cid:116) {d3,...,dn}. In the second
by asking that all irreducibles are weak primes, i.e., that the case, again by the hypothesis, there is i2 ↔ i1, hence by
(cid:70)
domain is algebraic with respect to weak primes. transitivity i2 ↔ i such that i2 (cid:118) d2 or i1 (cid:118) {d3,...,dn}.
Thuswefinallygettheexistenceofsomei(cid:48) ↔iandj ∈[1,n]
Definition 12 (weak prime algebraic domains) Let D be a such that i(cid:48) (cid:118) d . Recalling that d ∈ (cid:83)X, there is I ∈ X
j j
domain. A domain D is a weak prime algebraic domain (or
such that d ∈ I, hence ↓i ⊆ ↓d ⊆ I. Noting that i ↔ i
j j j j
simply weak prime domain) if for any d ∈ D it holds d =
implies that also in Idl(P) the irreducibles ↓i and ↓i are
(cid:70) j
(↓d∩wpr(D)).
interchangeable, i.e., ↓i↔↓i , we conclude.
j
In the same way as prime domains are domains where all We finally introduce a category of weak prime domains by
irreducibles are primes (see Proposition 2), we can provide a defining a notion of morphism.
characterisation of weak prime domains in terms of coinci-
Definition 13 (category of weak prime domains) A weak
dence between irreducibles and weak primes.
prime domain morphism f : D → D is a total function
1 2
Proposition 3 (weak prime domains, again) Let D be a such that
domain. It is a weak prime domain iff all irreducibles are 1) d (cid:22)d(cid:48) implies f(d )(cid:22)f(d(cid:48));
1 1 1 1
(cid:70) (cid:70)
weak primes. 2) for X ⊆D consistent, f( X )= f(X );
1 1 1 1
3) for d1,d(cid:48)1 ∈ D1 consistent, if d1(cid:117)d(cid:48)1 (cid:22) d1 then f(d1(cid:117) f(d1(cid:117)d(cid:48)1)=
d(cid:48)1)=f(d1)(cid:117)f(d(cid:48)1); f((e1(cid:117)d(cid:48)1)(cid:116)(f1(cid:117)d(cid:48)1))=
[Preservation of (cid:116)]
WedenotebywDomthecategoryofweakprimedomainsand f(e (cid:117)d(cid:48))(cid:116)f(f (cid:117)d(cid:48))=
1 1 1 1
their morphisms.
[Inductive hypothesis]
(f(e )(cid:117)f(d(cid:48)))(cid:116)(f(f )(cid:117)f(d(cid:48)))=
Compared with the notion of morphism for prime domains 1 1 1 1
in Definition 8, taken from [15], note that we still require the [Distributivity]
preservation of (cid:22) and (cid:116) of consistent sets (conditions (1) and (f(e )(cid:116)f(f ))(cid:117)f(d(cid:48))=
1 1 1
(2)). However, the third condition, i.e., preservation of (cid:117), is
[Preservation of (cid:116)]
weakened to preservation in some cases. General preservation
f(e (cid:116)f )(cid:117)f(d(cid:48))=
of meets is indeed not expected in the presence of fusions. 1 1 1
Consider e.g. the running example in Example 1 and another f(d1)(cid:117)f(d(cid:48)1)
es E(cid:48) = {c} with ∅ (cid:96) c and the morphism f : E → E(cid:48) If instead d is prime then note that if d (cid:118) d(cid:48) the thesis
1 1 1
thatforgetsaandb.Thenf({a,c})(cid:117)f({b,c})={c}(cid:117){c}= is immediate: by monotonicity f(d ) (cid:118) f(d(cid:48)). Thus f(d (cid:117)
1 1 1
{c}=(cid:54) f({a,c}(cid:117){b,c})=f(∅)=∅.Intuitively,thecondition d(cid:48)) = f(d ) = f(d )(cid:117)f(d(cid:48)) as desired. Therefore, let us
1 1 1 1
d (cid:117)d(cid:48) ≺d meansthatd(cid:48) includesthecomputationmodelled assume that d (cid:54)(cid:118)d(cid:48). In this case d (cid:117)d(cid:48) =p(d )(cid:117)d(cid:48), since
1 1 1 1 1 1 1 1 1 1
by d apart from a final step, hence d (cid:117)d(cid:48) coincides with the set of lower bounds of {d ,d(cid:48)} and of {p(d ),d(cid:48)} is the
1 1 1 1 1 1 1
d when such final step is removed. Since domain morphisms same. Observe that
1
preserve immediate precedence (i.e., single steps), also f(d )
1 p(d )=d (cid:117)(p(d )(cid:116)d(cid:48)) (1)
differs from f(d(cid:48)) for the execution of a final step and the 1 1 1 1
1
meet f(d )(cid:117)f(d(cid:48)) is f(d ) without such step, and thus it In fact, by distributivity, d (cid:117)(p(d )(cid:116)d(cid:48)) = (d (cid:117)p(d ))(cid:116)
1 1 1 1 1 1 1 1
coincides with f(d (cid:117)d(cid:48)). (d (cid:117)d(cid:48))=p(d )(cid:116)(p(d )(cid:117)d(cid:48))=p(d )
1 1 1 1 1 1 1 1
Therefore
In general we only have
f(d (cid:117)d(cid:48))=
1 1
f(p(d )(cid:117)d(cid:48))=
1 1
f((cid:100)X1)(cid:118)(cid:100)f(X1) [Inductive hypothesis]
f(p(d ))(cid:117)f(d(cid:48))=
In fact, for all x1 ∈ X1, we have (cid:100)X1 (cid:118) x1, hence 1 1
f((cid:100)X1) (cid:118) f(x1) and thus f((cid:100)X1) (cid:118) (cid:100)f(X1). Still, [Using (1)]
when restricted to prime domains, our notion of morphism f(d (cid:117)(p(d )(cid:116)d(cid:48)))(cid:117)f(d(cid:48))=
1 1 1 1
can be shown to boil down to the original one, i.e., the full
[By Definition 13(3)]
subcategory of wDom having prime domains as objects is
f(d )(cid:117)f(p(d )(cid:116)d(cid:48)))(cid:117)f(d(cid:48))=
pDom. 1 1 1 1
[Preservation of (cid:116)]
Lemma 7 (meet preservation for prime domains) Let D1, f(d1)(cid:117)f(d(cid:48)1)
D2 beprimedomainsandf :D1 →D2 aweakprimedomain asdesired.Thisextendstothemeetoffinitesetsofcompacts,
morphism. Then f((cid:100)X1)=(cid:100)f(X1). by associativity of (cid:117), and to infinite sets of compacts by
observing that, given an infinite set X, by finitariness we can
Proof: We first show that for d1,d(cid:48)1 ∈ K(D1), it holds identify a finite subset F ⊆X such that (cid:100)X =(cid:100)F.
that f(d (cid:117)d(cid:48))=f(d )(cid:117)f(d(cid:48)). We proceed by induction on
1 1 1 1 Proposition 4 The category of prime domains pDom is the
k =|↓d ∩pr(D)|.
1
full subcategory of wDom having prime domains as objects.
When k =0 we have d =⊥. Since f preserves joins, we
1
(cid:70) (cid:70) (cid:70) B. Connected es
have that f(⊥)=f( ∅)= f(∅)= ∅=⊥. Hence
Weshowthat,givenanes,itssetofconfigurations,ordered
by subset inclusion, is a weak prime domain. More precisely,
f(d (cid:117)d(cid:48))=f(⊥(cid:117)d(cid:48))=f(⊥)=⊥=⊥(cid:117)f(d(cid:48))=
1 1 1 1 since configurations arefinite sets, they representthe compact
f(⊥)(cid:117)f(d(cid:48))=f(d )(cid:117)f(d(cid:48)).
1 1 1 elements of the domain, which is thus obtained by ideal
completion. Moreover, the correspondence can be lifted to
Suppose now k > 0. We distinguish two subcases. If d
1 a functor. We also identify a subclass of es that we call
is not prime then, recalling that in a prime domain, primes
connectedesandthataretheexactcounterpartofweakprime
and irreducibles coincide, d is not irreducible and thus d =
1 1 domains (in the same way as prime es correspond to prime
e (cid:116)f with d (cid:54)= e ,f (cid:54)= ⊥. It is immediate to see that
1 1 1 1 1 algebraic domains).
|↓e ∩pr(D)|<k and |↓f ∩pr(D)|<k. Moreover, since
1 1
any prime algebraic domain is distributive we have d (cid:117)d(cid:48) = Definition 14 (poset of configurations of an es) Let E
1 1
(e (cid:116)f )(cid:117)d(cid:48) =(e (cid:117)d(cid:48))(cid:116)(f (cid:117)d(cid:48)). Summing up be an es. We denote by D(E) the ideal completion
1 1 1 1 1 1 1
Idl((cid:104)Conf(E),⊆(cid:105)). Given an es morphism f : E → E , 4) Let I = (cid:104)C ,e (cid:105) ∈ ir(D(E)) for j ∈ {1,2} be irre-
1 2 j j j
its image D(f) : D(E ) → D(E ) is defined as ducibles. Assume I ↔ I . By Lemma 4(3), observing
1 2 1 2
D(f)(C )={f(e ):e ∈C }. thatp(I )=C ,wemusthaveI ∪C =C ∪I ,namely
1 1 1 1 j j 1 2 1 2
C ∪{e }∪C =C ∪C ∪{e },fromwhichweconclude
Wefirstneedsometechnicalfacts,collectedinthefollowing 1 1 2 1 2 2
that it must be e = e , i.e., as desired I = (cid:104)C ,e(cid:105),
lemma. Recall that in the setting of unstable es we can have 1 2 j j
where e = e = e for j ∈ {1,2}. Additionally, I and
de,isCtin(cid:48)c(cid:96)tcoen,sainsdtenCt∪miCni(cid:48)m∪a{lee}naisblcinognssifsoternatn, weveenwtr.iWtehCen(cid:95)eCC(cid:96)(cid:48)0. I2 are consis1tent b2y Lemma 3, meaning that C1 (cid:95)e1C2.
0 e ∗ e For the converse, consider two irreducibles I1 =(cid:104)C1,e(cid:105)
We denote by (cid:95) the transitive closure of the relation (cid:95). e
and I = (cid:104)C ,e(cid:105), such that C (cid:95) C . Hence C (cid:96) e,
2 2 1 2 1 0
Lemma 8 Let (cid:104)E,(cid:96),Con(cid:105) be an es. Then C2 (cid:96)0 e and C = C1 ∪C2 ∪{e} is consistent. Since
I ,I ⊆ C, they are consistent in D(E). Moreover,
1) D(E) is a domain, K(D(E)) = Conf(E), join is union 1 2
and C (cid:22)C(cid:48) iff C =C(cid:48)∪{e} for some e∈E; p(I1) = C1, p(I2) = C2 and I1 ∪ C2 = I2 ∪ C1 =
C (cid:54)=C ∪C . Hence by Lemma 4(3), we have I ↔I ,
2) C ∈Conf(E) is irreducible iff C =C(cid:48)∪{e} and C(cid:48) (cid:96) 1 2 1 2
0
as desired.
e; it is denoted as I =(cid:104)C(cid:48),e(cid:105);
3) for C ∈ Conf(E), we have ir(C) = {(cid:104)C(cid:48),e(cid:48)(cid:105) | e(cid:48) ∈
C ∧ C(cid:48) ⊆C ∧ C(cid:48) (cid:96) e(cid:48)}; Concerning point 1, observe that the meet in the domain
0
4) for (cid:104)C ,e (cid:105),(cid:104)C ,e (cid:105) ∈ ir(D(E)), we have (cid:104)C ,e (cid:105) ↔ of configurations is C (cid:117)C(cid:48) = (cid:83){C(cid:48)(cid:48) ∈ Conf(E) | C(cid:48)(cid:48) ⊆
1 1 2 2 1 1
(cid:104)C ,e (cid:105) iff e=e =e and C (cid:95)e C . C ∧ C(cid:48)(cid:48) ⊆C(cid:48)},whichisusuallysmallerthantheintersection.
2 2 1 2 1 2
Forinstance,inFig.2,{a,c}(cid:117){b,c}=∅=(cid:54) {c}.Point2says
Proof:
that irreducibles are configurations of the form C ∪{e} that
1) Observe that the poset of configurations (cid:104)Conf(E),(cid:118) admits a secured execution in which the event e appears as
(cid:105) is finitary since configurations are finite and given the last one and cannot be switched with any other. In other
C,C(cid:48),C(cid:48)(cid:48) ∈ Conf(E), pairwise consistent, the join words, irreducibles are minimal histories of events. Point 3
C (cid:116)C(cid:48) is the union C (cid:116)C(cid:48), which is consistent with characterises the irreducibles in a configuration. According to
C(cid:48)(cid:48). The fact that D(E) is a domain thus follows by point 4, two irreducibles are interchangeable when they are
Lemma 6. different histories for the same event.
Concerningimmediateprecedence,letC,C(cid:48) ∈Conf(E).
If C(cid:48) = C ∪ {e} then clearly C ≺ C(cid:48), since the Proposition 5 Let E be an es. Then D(E) is a weak prime
order is subset inclusion. Vice versa, if C ≺ C(cid:48) by domain. Given a morphism f : E1 → E2, its image D(f) :
definition C ⊆ C(cid:48) and it must be |C(cid:48) \ C| = 1. In D(E1)→D(E2) is a weak prime domain morphism.
fact, let C = {e1,...,em} with (cid:104)e1,...,em(cid:105) a secured Proof: By Lemma 8 we know that D(E) is a domain.
execution and let C(cid:48) \ C = {e ,...,e }. Since
m+1 n In order to show that D(E) is a weak prime domain, we
enabling is monotone, there is a secured execution of C(cid:48)
exploit the characterisation in Proposition 3, i.e., we prove
of the kind (cid:104)e ,...,e ,e ,...,e (cid:105). Now, if it were
1 m m+1 n that all irreducibles are weak primes. Consider an irreducible,
|C(cid:48) \C| > 1 the set C(cid:48)(cid:48) = {e ,...,e ,e } would
1 m m+1 which,byLemma8(2),isoftheshapeI =(cid:104)C,e(cid:105)andsuppose
be a configuration, with C ⊂ C(cid:48)(cid:48) ⊂ C(cid:48), contradicting (cid:70)
that I ⊆ X for some X ⊆ K(D(E)). This means that, in
C ≺C(cid:48). particular,e∈(cid:70)X andthusthereisC(cid:48) ∈X suchthate∈C(cid:48).
In turn, we can consider a minimal enabling of e in C(cid:48), i.e.,
2) Let C ∈ Conf(E) be a configuration and assume that
C = C(cid:48) ∪{e} such that C(cid:48) (cid:96) e. If C = C ∪C for a minimal C(cid:48)(cid:48) ⊆ C(cid:48) such that C(cid:48)(cid:48) (cid:96)0 e, and we have that
0 1 2
I(cid:48)(cid:48) = (cid:104)C(cid:48)(cid:48),e(cid:105) is an irreducible I(cid:48)(cid:48) ⊆ C(cid:48). Since I and I(cid:48)(cid:48) are
C ,C ∈ Conf(E), then e must occur either in C or
1 2 1 (cid:70)
in C . If e∈C , since C is secured, there exists C(cid:48) ⊆ consistent, as they are both included in X, by Lemma 8(4),
C \2{e} such 1that C(cid:48) (cid:96)1e. Hence, by monotonicity1of I ↔I(cid:48)(cid:48).
1 1
enabling, C \{e}(cid:96)e. Since C(cid:48) (cid:96) e and C \{e}⊆C(cid:48)
1 0 1
it follows that C \{e}=C(cid:48) and thus C =C(cid:48). We next prove that given an es morphism f : E → E ,
1 1 1 2
Vice versa, let C ∈Conf(E) be irreducible. Consider a its image D(f) : D(E ) → D(E ) is a weak prime domain
1 2
secured execution (cid:104)e ,...,e (cid:105) of configuration C. Note morphism.
1 n
that for any k ∈ [1,n−1] it must be {e1,...,ek−1} (cid:54)(cid:96) • C1 (cid:22)C1(cid:48) implies D(f)(C1)(cid:22)D(f)(C1(cid:48))
e . Otherwise, if it were {e ,...,e } (cid:96) e for some
n 1 k−1 n
k ∈[1,n−1],wewouldhavethatC(cid:48) ={e1,...,ek,en} SinceD(f)(C )={f(d )|d ∈C }andbyLemma8(1)
i i i i
and C(cid:48)(cid:48) = {e1,...,en−1} are two proper subconfig- C (cid:22) C(cid:48) iff C(cid:48) = C ∪{e } for some event e , the
urations of C such that C = C(cid:48) ∪ C(cid:48)(cid:48), violating the 1 1 1 1 1 1
result follows immediately.
fact that C is irreducible. But this means exactly that
{e1,...,en−1}(cid:96)0 en, as desired. • for X1 ⊆ D(E1) consistent, D(f)((cid:70)X1) =
(cid:70)
D(f)(X )
3) Immediate. 1
Since D(f) takes the image as set and (cid:70) on consistent {a,c} {b,c} {a,c1} {b,c2}
sets is union, the result follows.
{a} {b} {a} {b}
• for C1,C1(cid:48) ∈ D(E1) consistent such that C1(cid:117)C1(cid:48) ≺ C1 ∅ ∅
it holds f(C (cid:117)C(cid:48))=f(C )(cid:117)f(C(cid:48))
1 1 1 1
Fig. 4: Non-connected es do not uniquely determine the
The condition C (cid:117) C(cid:48) ≺ C means that C (cid:117) C(cid:48) = domain of configurations.
1 1 1 1 1
C \{e } for some e ∈E . Clearly e (cid:54)∈C(cid:48), otherwise
1 1 1 1 1 1
also C ⊆C(cid:48) and thus C ∩C(cid:48) =C . Therefore in this
1 1 1 1 1
case, the meet coincides with intersection, C (cid:117)C(cid:48) = are different histories of the same event. In fact, note that we
1 1
C ∩C(cid:48) = C \{e }. Since for the events in C ∪C(cid:48), would get an isomorphic domain of configuration for the es
1 1 1 1 1 1
by definition of event structure morphism, f is injective, E(cid:48)(cid:48)witheventsE(cid:48)(cid:48) ={a,b,c1,c2}suchthata#bandminimal
we have that f(C1)∩f(C1(cid:48))=f(C1∩C1(cid:48)). As a general enablings are again ∅(cid:96)0 a, ∅(cid:96)0 b, {a}(cid:96)0 c1, and {b}(cid:96)0 c2.
fact,f(C )(cid:117)f(C(cid:48))⊆f(C )∩f(C(cid:48)).Therefore,putting
1 1 1 1
things together, we conclude C. From domains to es
f(C1)(cid:117)f(C1(cid:48))⊆f(C1)∩f(C1(cid:48))=f(C1∩C1(cid:48))= We show how to extract an es from a weak prime domain.
f(C1(cid:117)C1(cid:48)) As expected, events are equivalence classes of irreducibles,
Theconverseinequalityholdsinanydomain(asobserved wheretheequivalenceis(thetransitiveclosureof)interchange-
after Definition 13) and thus the result follows. ability.
In order to properly relate domains to the corresponding es
we need to prove some properties of irreducibles and of the
A special role is played by the subclass of connected es.
interchangeability relation in weak prime domains.
Definition 15 (connected es) AconnectedesisanesE such We already observed that the interchangeability relation in
that whenever C (cid:96)0 e and C(cid:48) (cid:96)0 e then C (cid:95)e ∗ C(cid:48). We denote generalisnottransitive.Weshowthatinweakprimedomains
by cES the full subcategory of ES having connected es as itistransitiveonconsistentirreducibles.Westartwithasimple
objects. technical lemma.
In words, different minimal enablings for the same event Lemma 9 Let D be a domain and i,i(cid:48),i(cid:48)(cid:48) ∈ir(D). If i↔i(cid:48),
must be pairwise connected by a chain of consistency. For i↔i(cid:48)(cid:48), and i(cid:48) (cid:118)i(cid:48)(cid:48) then i(cid:48) =i(cid:48)(cid:48).
instance, the es in Example 1 is a connected es. Only event
c has two minimal histories {a} (cid:96) c and {b} (cid:96) c and Proof: Assume i ↔ i(cid:48), i ↔ i(cid:48)(cid:48), and i(cid:48) (cid:118) i(cid:48)(cid:48) and suppose
0 0
obviously {a} (cid:95)c {b}. Clearly, prime es are also connected by absurdum that i(cid:48) (cid:54)=i(cid:48)(cid:48), hence i(cid:48) (cid:118)p(i(cid:48)(cid:48)). Then we have
es. More precisely, we have the following. i (cid:118)i(cid:116)p(i(cid:48))
=p(i)(cid:116)i(cid:48) [By i↔i(cid:48) and Lemma 4(3)]
Proposition 6 (connectedness, stability, primality) Let E
be an es. Then E is prime iff E is stable and connected. (cid:118)p(i)(cid:116)p(i(cid:48)(cid:48)) [Since i(cid:48) (cid:118)p(i(cid:48)(cid:48))]
Proof: The fact that a prime es is stable and connected Therefore p(i)(cid:116)p(i(cid:48)(cid:48))(cid:118)i(cid:116)p(i(cid:48)(cid:48))(cid:118)p(i)(cid:116)p(i(cid:48)(cid:48))(cid:116)p(i(cid:48)(cid:48))=
follows immediately from the definitions. Conversely, let E p(i)(cid:116)p(i(cid:48)(cid:48)). Hence we deduce i(cid:116)p(i(cid:48)(cid:48))=p(i)(cid:116)p(i(cid:48)(cid:48)), and
be stable and connected. We show that E is prime, i.e., each by Lemma 4(3) this contradicts i↔i(cid:48)(cid:48).
e∈E has a unique minimal enabling. Let C,C(cid:48) ∈Conf(E)
Lemma 10 (↔ transitive on consistent irreducibles) Let
be minimal enablings for e, i.e., C (cid:96) e and C (cid:96) e. Since
E is connected C (cid:95)e ∗ C(cid:48). Let C (cid:95)e C01 (cid:95)e ... (cid:95)e C0n (cid:95)e C(cid:48). Di↔bei(cid:48),aiw↔eaik(cid:48)(cid:48),parnimdei(cid:48)d,oi(cid:48)m(cid:48) acoinnsaisntdenit,.iT(cid:48),hie(cid:48)(cid:48)n∈i(cid:48) ↔ir(Di(cid:48)(cid:48)). such that
Then stability we get that C =C =...=C =C(cid:48).
1 n
The defining property of connected es allows one to recog- Proof: Since i ↔ i(cid:48) and i(cid:48) ↔ i(cid:48)(cid:48), by Lemma 3 i,i(cid:48) and
nise that two minimal histories are relative to the same event i(cid:48),i(cid:48)(cid:48) are consistent and by Lemma 4 p(i)(cid:116)i(cid:48) = i(cid:116)p(i(cid:48))
by only looking at the partially ordered structure and thus, as and p(i(cid:48))(cid:116)i(cid:48)(cid:48) = i(cid:48) (cid:116)p(i(cid:48)(cid:48)). Moreover i,i(cid:48)(cid:48) is consistent by
wewillsee,fromthedomainofconfigurationsofaconnected hypothesis and thus, by coherence, i,i(cid:48),i(cid:48)(cid:48) is consistent.
es we can recover an es isomorphic to the original one and Therefore i (cid:118) i (cid:116) p(i(cid:48)) (cid:116) p(i(cid:48)(cid:48)) = p(i) (cid:116) i(cid:48) (cid:116) p(i(cid:48)(cid:48)) =
vice versa (see Theorem 2). In general, this is not possible. p(i)(cid:116)p(i(cid:48))(cid:116)i(cid:48)(cid:48).SinceDisaweakprimedomain,irreducibles
For instance, consider the es E(cid:48) with events E(cid:48) = {a,b,c}, are weak primes and there must exist i1 ↔i such that either
and where a#b and minimal enablings are again ∅ (cid:96)0 a, i1 (cid:118) p(i) or i1 (cid:118) p(i(cid:48)) or i1 (cid:118) i(cid:48)(cid:48). The first possibility
∅ (cid:96)0 b, {a} (cid:96)0 c, and {b} (cid:96)0 c. Namely, event c has two i1 (cid:118) p(i) (cid:118) i contradicts Lemma 3, while the second one
minimal enablings, but differently from what happens in the i1 (cid:118)p(i(cid:48))(cid:118)i(cid:48) contradicts Lemma 9.
running example, these are not consistent, hence {a,b} (cid:54)(cid:96) c. Thereforeitmustbei (cid:118)i(cid:48)(cid:48).Wealsonotethatitcannotbe
1
Thedomainof configurationsisdepictedonthe leftofFig.4. i (cid:118) p(i(cid:48)(cid:48)), and thus, by Lemma 1, i(cid:48)(cid:48) = i ↔ i, as desired.
1 1
Intuitively,itisnotpossibletorecognisethat{a,c}and{b,c} Otherwise, if i (cid:118)p(i ), by Lemma 4(2) we would have
1 2
i(cid:116)p(i)(cid:116)p(i(cid:48))(cid:116)p(i(cid:48)(cid:48)) = i1(cid:116)p(i)(cid:116)p(i(cid:48))(cid:116)p(i(cid:48)(cid:48)) d≺d(cid:48). Then for any i∈δ(d(cid:48),d)
= p(i)(cid:116)p(i(cid:48))(cid:116)p(i(cid:48)(cid:48))
1) d(cid:48) =d(cid:116)i;
In turn, the above equality could be used to prove that Moreover, for all i,i(cid:48) minimal in δ(d(cid:48),d)
i(cid:48) (cid:118) i(cid:48)(cid:116)p(i)(cid:116)p(i(cid:48))(cid:116)p(i(cid:48)(cid:48)) 2) there is no i(cid:48)(cid:48) ∈ir(d) such that i↔i(cid:48)(cid:48);
= i(cid:116)p(i)(cid:116)p(i(cid:48))(cid:116)p(i(cid:48)(cid:48)) 3) i↔i(cid:48).
= p(i)(cid:116)p(i(cid:48))(cid:116)p(i(cid:48)(cid:48))
Proof:
Again,bythefactthatDisaweakprimedomain,thereshould 1) Let i ∈ δ(d(cid:48),d). Then d (cid:118) d(cid:116)i (cid:118) d(cid:48). It follows that
exist i(cid:48)1 ↔i(cid:48) such that i(cid:48)1 (cid:118)p(i) or i(cid:48)1 (cid:118)p(i(cid:48)) or i(cid:48)1 (cid:118)p(i(cid:48)(cid:48)). either d(cid:116)i = d or d(cid:116)i = d(cid:48). The first possibility can
Note that i(cid:48) (cid:118) p(i(cid:48)) (cid:64) i(cid:48) contradicts Lemma 3. Moreover,
1 be excluded for the fact that it would imply i(cid:118)d, while
recalling that i(cid:48) ↔ i and i(cid:48) ↔ i(cid:48), i(cid:48) (cid:118) p(i) (cid:64) i contradicts
1 1 we know that i(cid:54)∈ir(d). Hence we get the thesis.
Lemma 9. The same applies to the case i(cid:48)1 (cid:118) p(i(cid:48)(cid:48)) (cid:64) i(cid:48)(cid:48). 2) Let i ∈ δ(d(cid:48),d) minimal. Therefore p(i) (cid:118) d. If there
Hence we conclude. werei(cid:48)(cid:48) ∈ir(d)suchthati↔i(cid:48)(cid:48) then,sincealsop(i(cid:48)(cid:48))(cid:118)
Domainsareirreduciblealgebraic,henceanyelementisde- d, by Lemma 4(2) we would get d = d(cid:116)i(cid:48)(cid:48) = d(cid:116)i =
termined by the irreducibles under it. The difference between d(cid:48), the last equality following from point (1). But this
twoelementsisthussomehowcapturedbytheirreduciblesthat contradicts the hypothesis that d≺d(cid:48).
areunderoneelementandnotundertheother.Thismotivates 3) Let i,i(cid:48) ∈ δ(d(cid:48),d) be minimal irreducibles. By (1) we
the following definition. have d(cid:48) = d (cid:116) i. Therefore i(cid:48) ∈ δ(d(cid:116)i,d) minimal.
Definition 16 (irreducible difference) Let D be a domain Therefore i↔i(cid:48) by Lemma 11(1).
and d,d(cid:48) ∈ K(D) such that d (cid:118) d(cid:48). Then we define
δ(d(cid:48),d)=ir(d(cid:48))\ir(d). Wenextshowthatchainsofimmediateprecedencearegen-
eratedinessentiallyauniquewaybysequencesofirreducibles.
The immediate precedence relation intuitively relates do-
main elements corresponding to configurations which differ Lemma 13 (chains) Let D be a weak prime domain, d ∈
for the execution of a single event. We next study how this K(D) and ir(d) = {i ,...,i } such that the sequence
1 n
relates to irreducibles. i ,...,i is compatible with the order (i.e., for all h,k
1 n
if i (cid:118) i then h ≤ k). If we let d = (cid:70)k i for
Lemma 11 (immediate precedence and irreducibles/1) h k k h=1 h
k ∈{1,...,n} we have
Let D be a weak prime domain, d ∈ K(D), and i ∈ ir(D)
such that d, i are consistent and p(i)(cid:118)d. Then ⊥=d (cid:22)d (cid:22)...(cid:22)d =d
0 1 n
1) for all i(cid:48) ∈δ(d(cid:116)i,d) minimal, it holds i↔i(cid:48);
Vice versa, given a chain ⊥ = d ≺ d ≺ ... ≺ d and
0 1 n
2) d(cid:22)d(cid:116)i.
taking i ∈δ(d ,d ) minimal for h∈{1,...,n} we have
h h h−1
Proof: d =(cid:70)n i and ∀i∈ir(d ). ∃h∈[1,n]. i↔i .
n h=1 h n h
1) First note that if d=d(cid:116)i, hence ir(d(cid:116)i)=ir(d) and
Therefore [{i ,...,i }] =[ir(d )] .
the property trivially holds. Assume d (cid:54)= d(cid:116)i and take 1 n ↔∗ n ↔∗
i(cid:48) ∈ δ(d(cid:116)i,d) minimal. Note that minimality implies Proof: For the first part, observe that for k ∈{1,...,n}
that p(i(cid:48))(cid:118)d. From i(cid:48) (cid:118)d(cid:116)i, since D is a weak prime we have that
domainandthusirreduciblesareweakprimes,theremust
p(i )(cid:118)d
be i(cid:48)(cid:48) ∈ ir(D), i(cid:48)(cid:48) ↔ i(cid:48) such that i(cid:48)(cid:48) (cid:118) d or i(cid:48)(cid:48) (cid:118) i. We k k−1
firstnotethatitcannotbei(cid:48)(cid:48) (cid:118)d,otherwised=d(cid:116)i(cid:48)(cid:48) = Infact,recallingthatir(i )⊆ir(d),wehavethatirreducibles
k
d(cid:116)i(cid:48),thelastequalitymotivatedbyLemma4(2).Thuswe in ir(p(i )) = ir(i )\{i }, which are smaller than i , must
k k k k
would have i(cid:48) (cid:118) d, contradicting the hypotheses. Hence occur before in the list hence
it must be i(cid:48)(cid:48) (cid:118)i. This, in turn, by the minimality of i in
δ(d(cid:116)i,d) implies i=i(cid:48)(cid:48), hence i=i(cid:48)(cid:48) ↔i(cid:48), as desired. ir(p(ik))=ir(ik)\{ik}⊆{i1,...,ik−1}.
2) Consider d(cid:48) such that d ≺ d(cid:48) (cid:118) d(cid:116)i, and let us prove Therefore p(i ) = (cid:70)ir(p(i )) (cid:118) (cid:70){i ,...,i } = d .
k k 1 k−1 k−1
thatd(cid:48) =d(cid:116)i.Sinced≺d(cid:48),henced(cid:54)=d(cid:48),weknowthat Thus we use Lemma 11(2) to infer d (cid:22)d (cid:116)i =d .
k−1 k−1 k k
δ(d(cid:48),d) is not empty. Take a minimal i(cid:48) ∈δ(d(cid:48),d). Thus
i(cid:48) is minimal also in δ(d(cid:116)i,d), and thus, by point (1), For the second part, we proceed by induction on n.
i↔i(cid:48).Byminimalityofi(cid:48)wededucealsothatp(i(cid:48))(cid:118)d. (cid:70)
• (n = 0) Note that d0 = ∅ = ⊥ and ir(⊥) = ∅, hence
Since also p(i) (cid:118) d by hypothesis, using Lemma 4(2),
the thesis trivially holds.
we have d(cid:116)i=d(cid:116)i(cid:48). Observing that d(cid:116)i(cid:48) (cid:118)d(cid:48) (cid:118)d(cid:116)i
• (n>0) By induction hypothesis
we conclude that d(cid:48) =d(cid:116)i, as desired. d =(cid:70)n−1i and
n−1 h=1 h
∀i∈ir(d ). ∃h∈[1,n−1]. i↔i .
n−1 h
Lemma 12 (immediate precedence and irreducibles/2) Sincebyconstructioni ∈δ(d ,d ),byLemma12(1)
n n n−1
Let D be a weak prime domain and d,d(cid:48) ∈ D such that we deduce
